QuestionRewrite the given equations as quadratic equations in the form $a x^{2}+b x+c=0$ .

Studdy Solution

STEP 1

Assumptions1. All the given equations are quadratic equations. . The standard form of a quadratic equation is $a x^{}+b x+c=0$ .
3. We need to express each equation in the standard form.

STEP 2

Let's start with the first equation $(2 x+1)(x-1)=0$ . We can express this equation in standard form by expanding the expression on the left-hand side.
$2x^2 -2x + x -1 =0$

STEP 3

Combine like terms to simplify the equation.
$2x^2 - x -1 =0$

STEP 4

Now, let's move to the second equation $3 x(x+1)=(x-2)^{2}$ . We can express this equation in standard form by expanding both expressions on the left-hand side and the right-hand side.
$3x^2 +3x = x^2 -4x +4$

STEP 5

Rearrange the equation to bring all terms to one side.
$3x^2 - x^2 +3x +4x -4 =0$

STEP 6

Combine like terms to simplify the equation.
$2x^2 +x -4 =0$

STEP 7

Now, let's move to the third equation $(x-4)(2 x-5)=0$ . We can express this equation in standard form by expanding the expression on the left-hand side.
$2x^2 -5x -x +20 =0$

STEP 8

Combine like terms to simplify the equation.
$2x^2 -13x +20 =0$

STEP 9

Now, let's move to the fourth equation $(2 x-5)(x+)=3 x-2$ . We can express this equation in standard form by expanding the expression on the left-hand side and moving the terms on the right-hand side to the left-hand side.
$2x^2 +2x -5x -5 -3x +2 =$

STEP 10

Combine like terms to simplify the equation.
$2x^2 -6x -3 =0$

STEP 11

Now, let's move to the fifth equation $x(x+)=x+$ . We can express this equation in standard form by expanding the expression on the left-hand side and moving the term on the right-hand side to the left-hand side.
$x^ + x - x - =0$

STEP 12

Combine like terms to simplify the equation.
$x^2 -2 =0$

STEP 13

Now, let's move to the sixth equation $(3 x-)^{2}=x+$ . We can express this equation in standard form by expanding the expression on the left-hand side and moving the term on the right-hand side to the left-hand side.
$9x^2 -6x + - x - =0$

STEP 14

Combine like terms to simplify the equation.
$9x^2 -7x =0$

STEP 15

Now, let's move to the seventh equation $5 x=2+(x-)(x+2)$ . We can express this equation in standard form by expanding the expression on the right-hand side and moving the term on the left-hand side to the right-hand side.
$2 + x^2 +2x - x -5 =0$

STEP 16

Combine like terms to simplify the equation.
$x^2 + x -3 =0$

STEP 17

Now, let's move to the eighth equation $6 x(3 x+2)=2 x(10 x-)$ . We can express this equation in standard form by expanding both expressions on the left-hand side and the right-hand side, and then moving all terms to one side.
$x^2 +12x -20x^2 +2x =0$

STEP 18

Combine like terms to simplify the equation.
$-2x^2 -10x =0$

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